Wigner-Smith Time-delay Matrix Research Articles

Wigner–Smith Time Delay Matrix for Electromagnetics: Systems With Material Dispersion and Losses

The Wigner-Smith (WS) time delay matrix relates a system’s scattering matrix to its frequency derivative and gives rise to so-called WS modes that experience well-defined group delays when interacting with the system. For systems composed of nondispersive and lossless materials, the WS time delay matrix previously was shown to consist of volume integrals of energy-like densities plus correction terms that account for the guiding, scattering, or radiating characteristics of the system. This study extends the use of the WS time delay matrix to systems composed of dispersive and lossy materials. Specifically, it shows that such systems’ WS time delay matrix can be expressed by augmenting the previously derived expressions with terms that account for the dispersive and lossy nature of the system, followed by a transformation that disentangles effects of losses from time delays. Analytical and numerical examples demonstrate the new formulation once again allows for the construction of frequency stable WS modes that experience well-defined group delays upon interacting with a system.

Wigner-Smith time delay matrix for acoustic scattering: Theory and phenomenology.

The Wigner-Smith (WS) time delay matrix relates a lossless system's scattering matrix to its frequency derivative. First proposed in the realm of quantum mechanics to characterize time delays experienced by particles during a collision, this article extends the use of WS time delay techniques to acoustic scattering problems governed by the Helmholtz equation. Expression for the entries of the WS time delay matrix involving renormalized volume integrals of energy densities are derived, and shown to hold true, independent of the scatterer's geometry, boundary condition (sound-soft or sound-hard), and excitation. Numerical examples show that the eigenmodes of the WS time delay matrix describe distinct scattering phenomena characterized by well-defined time delays.

Wigner–Smith Time-Delay Matrix for Electromagnetics: Guiding and Periodic Systems With Evanescent Modes

The Wigner-Smith (WS) time delay matrix relates an electromagnetic system’s scattering matrix and its frequency derivative. Previous work showed that the entries of WS time delay matrices of systems excited by propagating waves consist of volume integrals of energy-like field quantities. This paper introduces a generalized WS relationship that applies to systems excited by mixtures of propagating and evanescent fields. Just like its predecessor, the generalized WS relationship allows for the identification of so-called WS modes that interact with the system with well-defined time delays. Furthermore, a technique is developed to compute the WS time delay matrix of a composite system from the WS time delay matrices of its subsystems. Numerical examples demonstrate the usefulness of the generalized WS method when characterizing time delays experienced by fields interacting with guiding and periodic structures that have ports supporting evanescent modes.

Semiclassical calculation of time delay statistics in chaotic quantum scattering

We present a semiclassical calculation, based on classical action correlations implemented by means of a matrix integral, of all moments of the Wigner–Smith time delay matrix, Q, in the context of quantum scattering through systems with chaotic dynamics. Our results are valid for broken time reversal symmetry and depend only on the classical dwell time and the number of open channels, M, which is arbitrary. Agreement with corresponding random matrix theory reduces to an identity involving some combinatorial concepts, which can be proved in special cases.

Integer moments of complex Wishart matrices and Hurwitz numbers

We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained factorisations in the symmetric group. The two expansions can be compared and combined with a duality relation proved in \[F. D. Cunden, F. Mezzadri, N. O’Connell, and N. J. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys. 369 (2019), no. 3, 1091–1145] to obtain: i) a combinatorial proof of the reflection formula between moments of LUE and inverse LUE at genus zero and, ii) a new functional relation between the generating functions of monotone and strictly monotone Hurwitz numbers. The main result resolves the integrality conjecture formulated in \[F. D. Cunden, F. Mezzadri, N. J. Simm, and P. Vivo, Correlators for the Wigner–Smith time-delay matrix of chaotic cavities, J. Phys. A 49 (2016), no. 18, 18LT01, 20 pp] on the time-delay cumulants in quantum chaotic transport. The precise combinatorial description of the cumulants given here may cast new light on the concordance between random matrix and semiclassical theories.

Wigner–Smith Time Delay Matrix for Electromagnetics: Computational Aspects for Radiation and Scattering Analysis

The WS time delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative, and enables the synthesis of modes that experience well-defined group delays when interacting with the system. The elements of the WS time delay matrix for surface scatterers and antennas comprise renormalized energy-like volume integrals involving electric and magnetic fields that arise when exciting the system via its ports. Here, direct and indirect methods for computing the WS time delay matrix are presented. The direct method evaluates the energy-like volume integrals using surface integral operators that act on the incident electric fields and current densities for all excitations characterizing the scattering matrix. The indirect method accomplishes the same task by computing scattering parameters and their frequency derivatives. Both methods are computationally efficient and readily integrated into existing surface integral equation codes. The proposed techniques facilitate the evaluation of frequency derivatives of antenna impedances, antenna patterns, and scatterer radar cross sections in terms of renormalized field energies derived from a single frequency characterization of the system.

Wigner–Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity

We consider a multichannel wire with a disordered region of length L and a reflecting boundary. The reflection of a wave of frequency ω is described by the scattering matrix , encoding the probability amplitudes to be scattered from one channel to another. The Wigner–Smith time delay matrix is another important matrix, which encodes temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices, (with in the presence of time reversal symmetry), and introduce a novel symmetrisation procedure for the Wigner–Smith matrix: , where k is the wave vector and v the group velocity. We demonstrate that can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires, L → ∞, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for ’s eigenvalues of Brouwer and Beenakker (2001 Physica E 9 463). For finite length L, the exponential functional representation is used to calculate the first moments , and . Finally we derive a partial differential equation for the resolvent in the large N limit.

Wigner–Smith Time-Delay Matrix for Electromagnetics: Theory and Phenomenology

Wigner-Smith (WS) time delay concepts have been used extensively in quantum mechanics to characterize delays experienced by particles interacting with a potential well. This paper formally extends WS time delay theory to Maxwell's equations and explores its potential applications in electromagnetics. The WS time delay matrix relates a lossless and reciprocal system's scattering matrix to its frequency derivative and allows for the construction of modes that experience well-defined group delays when interacting with the system. The matrix' entries for guiding, scattering, and radiating systems are energy-like overlap integrals of the electric and/or magnetic fields that arise upon excitation of the system via its ports. The WS time delay matrix has numerous applications in electromagnetics, including the characterization of group delays in multiport systems, the description of electromagnetic fields in terms of elementary scattering processes, and the characterization of frequency sensitivities of fields and multiport antenna impedance matrices.

Distribution of the Wigner–Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases

Within the random matrix theory approach to quantum scattering, we derive the distribution of the Wigner–Smith time delay matrix for a chaotic cavity with uniform absorption, coupled via N perfect channels. In the unitary class we obtain a compact expression for the distribution of the full matrix in terms of a matrix integral. In the other symmetry classes we derive the joint distribution of the eigenvalues. We show how the large N properties of this distribution can be analysed in terms of two interacting Coulomb gases living on two different supports. As an application of our results, we study the statistical properties of the Wigner time delay in the presence of absorption.

Wigner–Smith time-delay matrix in chaotic cavities with non-ideal contacts

We consider wave propagation in a complex structure coupled to a finite number N of scattering channels, such as chaotic cavities or quantum dots with external leads. Temporal aspects of the scattering process are analysed through the concept of time delays, related to the energy (or frequency) derivative of the scattering matrix . We develop a random matrix approach to study the statistical properties of the symmetrised Wigner–Smith time-delay matrix , and obtain the joint distribution of and for the system with non-ideal contacts, characterised by a finite transmission probability (per channel) . We derive two representations of the distribution of in terms of matrix integrals specified by the Dyson symmetry index (the general case of unequally coupled channels is also discussed). We apply this to the Wigner time delay , which is an important quantity providing the density of states of the open system. Using the obtained results, we determine the distribution of the Wigner time delay in the weak coupling limit and identify the following three regimes. (i) The large deviations at small times (measured in units of the Heisenberg time) are characterised by the limiting behaviour for . (ii) The distribution shows the universal behaviour in some intermediate range . (iii) It has a power law decay for large .

Scattering Approach to Anderson Localization

We develop a novel approach to the Anderson localization problem in a d-dimensional disordered sample of dimension L×M^{d-1}. Attaching a perfect lead with the cross section M^{d-1} to one side of the sample, we derive evolution equations for the scattering matrix and the Wigner-Smith time delay matrix as a function of L. Using them one obtains the Fokker-Planck equation for the distribution of the proper delay times and the evolution equation for their density at weak disorder. The latter can be mapped onto a nonlinear partial differential equation of the Burgers type, for which a complete analytical solution for arbitrary L is constructed. Analyzing the solution for a cubic sample with M=L in the limit L→∞, we find that for d<2 the solution tends to the localized fixed point, while for d>2 to the metallic fixed point, and provide explicit results for the density of the delay times in these two limits.

Spatiotemporal Control of Light Transmission through a Multimode Fiber with Strong Mode Coupling.

We experimentally generate and characterize eigenstates of the Wigner-Smith time-delay matrix, called principal modes, in a multimode fiber with strong mode coupling. The unique spectral and temporal properties of principal modes enable global control of temporal dynamics of optical pulses transmitted through the fiber, despite random mode mixing. Our analysis reveals that well-defined delay times of the eigenstates are formed by multipath interference, which can be effectively manipulated by spatial degrees of freedom of input wave fronts. This study is essential to controlling dynamics of wave scattering, paving the way for coherent control of pulse propagation through complex media.

Correlators for the Wigner–Smith time-delay matrix of chaotic cavities

We study the Wigner–Smith time-delay matrix Q of a ballistic quantum dot supporting N scattering channels. We compute the v-point correlators of the power traces for arbitrary at leading order for large N using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the 's are integer-valued at leading order in N and include a MATHEMATICA code that computes their generating functions recursively.

Wigner time delay and related concepts: Application to transport in coherent conductors

The concepts of Wigner time delay and Wigner–Smith matrix allow us to characterise temporal aspects of a quantum scattering process. The paper reviews the statistical properties of the Wigner time delay for disordered systems; the case of disorder in 1D with a chiral symmetry is discussed and the relation with exponential functionals of the Brownian motion is underlined. Another approach for the analysis of time delay statistics is the random matrix approach, from which we review few results. As a practical illustration, we briefly outline a theory of non-linear transport and AC transport developed by Büttiker and coworkers, where the concept of Wigner–Smith time delay matrix is a central piece allowing us to describe screening properties in out-of-equilibrium coherent conductors.

Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities.

We derive the joint distribution of the moments TrQ(κ) (κ≥1) of the Wigner-Smith matrix for a chaotic cavity supporting a large number of scattering channels n. This distribution turns out to be asymptotically Gaussian, and we compute explicitly averages and covariances. The results are in a compact form and have been verified numerically. The general methodology of proof and computations has a wide range of applications.

Effect of chiral symmetry on chaotic scattering from Majorana zero modes.

In many of the experimental systems that may host Majorana zero modes, a so-called chiral symmetry exists that protects overlapping zero modes from splitting up. This symmetry is operative in a superconducting nanowire that is narrower than the spin-orbit scattering length, and at the Dirac point of a superconductor-topological insulator heterostructure. Here we show that chiral symmetry strongly modifies the dynamical and spectral properties of a chaotic scatterer, even if it binds only a single zero mode. These properties are quantified by the Wigner-Smith time-delay matrix Q=-iℏS^{†}dS/dE, the Hermitian energy derivative of the scattering matrix, related to the density of states by ρ=(2πℏ)^{-1}TrQ. We compute the probability distribution of Q and ρ, dependent on the number ν of Majorana zero modes, in the chiral ensembles of random-matrix theory. Chiral symmetry is essential for a significant ν dependence.

Generating Particlelike Scattering States in Wave Transport

We introduce a procedure to generate scattering states which display trajectorylike wave function patterns in wave transport through complex scatterers. These deterministic scattering states feature the dual property of being eigenstates to the Wigner-Smith time-delay matrix Q and to the transmission matrix t(†)t with classical (noiseless) transmission eigenvalues close to 0 or 1. Our procedure to create such beamlike states is based solely on the scattering matrix and successfully tested numerically for regular, chaotic, and disordered cavities. These results pave the way for the experimental realization of highly collimated wave fronts in transport through complex media with possible applications such as secure and low-power communication.

Density of proper delay times in chaotic and integrable quantum billiards.

We calculate the density P(tau) of the eigenvalues of the Wigner-Smith time delay matrix for two-dimensional rectangular and circular billiards with one opening. For long times, the density of these so-called "proper delay times" decays algebraically, in contradistinction to chaotic quantum billiards for which P(tau) exhibits a long-time cutoff.

Distribution of the quantum mechanical time-delay matrix for a chaotic cavity

We calculate the joint probability distribution of the Wigner-Smith time-delay matrix Q=−iℏS −1∂S/∂ε and the scattering matrix S for scattering from a chaotic cavity with ideal point contacts. To this end we prove a conjecture by Wigner about the unitary invariance property of the distribution functional P[S(ε)] of energy-dependent scattering matrices S(ε). The distribution of the inverse of the eigenvalues τ1,…,τ N of Q is found to be the Laguerre ensemble from random-matrix theory. The eigenvalue density ρ(τ) is computed using the method of orthogonal polynomials. This general theory has applications to the thermopower, magnetoconductance, and capacitance of a quantum dot.

Distribution of Parametric Conductance Derivatives of a Quantum Dot

The conductance $G$ of a quantum dot with single-mode ballistic point contacts depends sensitively on external parameters $X$, such as gate voltage and magnetic field. We calculate the joint distribution of $G$ and $\mathrm{dG}/\mathrm{dX}$ by relating it to the distribution of the Wigner-Smith time-delay matrix of a chaotic system. The distribution of $\mathrm{dG}/\mathrm{dX}$ has a singularity at zero and algebraic tails. While $G$ and $\mathrm{dG}/\mathrm{dX}$ are correlated, the ratio of $\mathrm{dG}/\mathrm{dX}$ and $\sqrt{G(1\ensuremath{-}G)}$ is independent of $G$. Coulomb interactions change the distribution of $\mathrm{dG}/\mathrm{dX}$, by inducing a transition from the grand-canonical to the canonical ensemble. All these predictions can be tested in semiconductor microstructures or microwave cavities.